Submodular Maximization with Matroid and Packing Constraints in Parallel
Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing(2018)
摘要
We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a 1-1/e-ϵ approximation for monotone functions and a 1/e-ϵ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is O(log^2n/ϵ^3), which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a 1/e-ϵ approximation using O(log(n/ϵ) log(1/ϵ) log(n+m)/ ϵ^2) parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a 1-1/e-ϵ approximation in O(log(n/ϵ)log(m)/ϵ^2) parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.
更多查看译文
关键词
DR-submodular maximization, matroid, packing, parallel complexity
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要